**1. Introduction**

In recent years, there has been intense research about ring resonators (RRs) as the building blocks for various photonic applications such as optical switches, wavelength multiplexers, routers, optical delay lines, and optical sensors [1–3]. In the literature, microring resonators with high-quality factors (Q) are required for enhanced nonlinear effects, low threshold lasing, and sensing applications. Almost all of the proposed microring resonator structures used directional couplers as coupling elements. It was shown that such devices are very sensitive to fabrication tolerance [4]. However, a very high Q is undesirable for high-speed signal processing since it can significantly limit the operational bandwidth of the system. In addition, in microring resonator structure, the coupling ratio and loss must be matched so that the ring operates near the critical coupling to achieve a high extinction ratio [5]. To obtain a high bandwidth, a solution is to use a directional coupler with a small gap between two waveguides in order to increase the coupling coefficient. However, this causes large excess mode conversion losses, limiting the flexibility of this approach [6].

Another type of coupler, namely, the 2 � 2 multimode interference (MMI) coupler, has been employed in ring resonators [7–10]. MMI couplers have been shown to have relaxed fabrication requirements and are less sensitive to the wavelength or polarization variations [11, 12]. In recent years, we have presented some microring resonators based on silicon waveguides using 2 � 2, 3 � 3 MMIs for the first time [4, 13–15]. It showed that the proposed devices have good performance compared with structures based on directional couplers.

In a single ring resonator, control of the critical coupling is an important requirement [16–18]. In the literature, the Mach-Zehnder (MZI) configuration is used for this purpose [17]. The key physical mechanism they rely on is the plasma dispersion effect or thermo-optic effect. Another approaches used to create phase shifter are based on silicon-organic hybrid slot waveguides [19] and BTO-Si slot waveguides [20]. The major drawbacks of these schemes are relatively large dimensions. In addition, plasma dispersion can only induce a small variation of the refractive index; the long length of the phase shifter is required. In this chapter, we use the graphene-silicon waveguide (GSW) for phase shifter and controlling the coupling coefficient.

It is noted that in the literature, the MZI configuration for critical coupling control is based on two 3 dB 2 � 2 directional couplers or 2 � 2 MMI coupler. In this chapter, we do not need to use the MZI for controlling the critical coupling of microring resonators, but we present a new way of achieving the critical coupling based on architecture itself. Our approach has advantages of compact size and ease of fabrication with the current CMOS circuit.

In addition, we use the graphene-silicon waveguide for the phase control. Graphene is a single-sheet carbon atom in a hexagonal lattice [21]. Graphene has some potential properties for optical devices. Graphene is a 2-D single-layer carbon atoms arranged in a hexagonal lattice that has raised considerable interest in recent years due to its remarkable optical and electronic properties. For example, graphene has a much higher electron mobility than silicon [22–24]. In particular, it has a linear dispersion relationship in the so-called Dirac points where electrons behave as fermions with zero mass. As a result, we can design optical switches or modulators based on this property. Graphene can also absorb light over a broad-frequency range, so this enables high-speed applications. The density of states of carriers near the Dirac point is low, and the Fermi energy can be tuned significantly with relatively low applied voltage. The Fermi-level tuning changes the refractive index of the graphene. Therefore, the graphene sheet integrated with optical waveguide such as silicon on insulator (SOI) waveguide can provide the possibilities of programmable in optoelectronics.

### **2. Microring resonator based on multimode waveguides**

#### **2.1 Device structure**

A new optical microring resonator based on only one multimode waveguide with four ports is shown in **Figure 1**.

**Figure 2(a)** shows the single-mode waveguide profile. We use silicon on insulator waveguide with a width of 500 nm and height of 250 nm for input and output waveguides. For a multimode waveguide, we use a wider width of *WMMI* ¼ 6*μm*. The field profile of the fundamental mode of the SOI waveguide is shown in **Figure 2(b)**. The refractive indices of silicon and silicon oxide used in our simulations are *nSi* ¼ 3*:*45, *nSiO*<sup>2</sup> ¼ 1*:*45. The field profiles of the fundamental mode and the first-order mode of the multimode waveguide are shown in **Figure 2(c)** and **(d)**.

In this structure, we use a bent waveguide to connect input port 3 to port 4 as a ring resonator waveguide. Because port 3 is very near to port 4, the bent waveguide radius is relatively small. Therefore, our structure can provide a very high free spectral range

*(a) SOI waveguide structure, (b) field profile of the single-mode SOI waveguide, (c) fundamental mode of the*

In the next section, we show that our structure can act like a microring resonator. In order to control the critical coupling, we use graphene integrated with silicon

(FSR), which is suitable for high-speed communications.

*multimode waveguide, and (d) the first-order mode of the multimode waveguide.*

*Microring resonator based on only one multimode waveguide structure.*

*DOI: http://dx.doi.org/10.5772/intechopen.92210*

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon…*

**Figure 1.**

**Figure 2.**

**51**

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon… DOI: http://dx.doi.org/10.5772/intechopen.92210*

**Figure 1.** *Microring resonator based on only one multimode waveguide structure.*

#### **Figure 2.**

Another type of coupler, namely, the 2 � 2 multimode interference (MMI) coupler, has been employed in ring resonators [7–10]. MMI couplers have been shown to have relaxed fabrication requirements and are less sensitive to the wavelength or polarization variations [11, 12]. In recent years, we have presented some microring resonators based on silicon waveguides using 2 � 2, 3 � 3 MMIs for the first time [4, 13–15]. It showed that the proposed devices have good performance

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

In a single ring resonator, control of the critical coupling is an important requirement [16–18]. In the literature, the Mach-Zehnder (MZI) configuration is used for this purpose [17]. The key physical mechanism they rely on is the plasma dispersion effect or thermo-optic effect. Another approaches used to create phase shifter are based on silicon-organic hybrid slot waveguides [19] and BTO-Si slot waveguides [20]. The major drawbacks of these schemes are relatively large dimensions. In addition, plasma dispersion can only induce a small variation of the refractive index; the long length of the phase shifter is required. In this chapter, we use the graphene-silicon waveguide (GSW) for phase shifter and controlling the

It is noted that in the literature, the MZI configuration for critical coupling control is based on two 3 dB 2 � 2 directional couplers or 2 � 2 MMI coupler. In this chapter, we do not need to use the MZI for controlling the critical coupling of microring resonators, but we present a new way of achieving the critical coupling based on architecture itself. Our approach has advantages of compact size and ease

In addition, we use the graphene-silicon waveguide for the phase control. Graphene is a single-sheet carbon atom in a hexagonal lattice [21]. Graphene has some potential properties for optical devices. Graphene is a 2-D single-layer carbon atoms arranged in a hexagonal lattice that has raised considerable interest in recent years due to its remarkable optical and electronic properties. For example, graphene has a much higher electron mobility than silicon [22–24]. In particular, it has a linear dispersion relationship in the so-called Dirac points where electrons behave as fermions with zero mass. As a result, we can design optical switches or modulators based on this property. Graphene can also absorb light over a broad-frequency range, so this enables high-speed applications. The density of states of carriers near the Dirac point is low, and the Fermi energy can be tuned significantly with relatively low applied voltage. The Fermi-level tuning changes the refractive index of the graphene. Therefore, the graphene sheet integrated with optical waveguide such as silicon on insulator (SOI) waveguide can provide the possibilities of program-

**2. Microring resonator based on multimode waveguides**

A new optical microring resonator based on only one multimode waveguide with

**Figure 2(a)** shows the single-mode waveguide profile. We use silicon on insulator waveguide with a width of 500 nm and height of 250 nm for input and output waveguides. For a multimode waveguide, we use a wider width of *WMMI* ¼ 6*μm*. The field profile of the fundamental mode of the SOI waveguide is shown in **Figure 2(b)**. The refractive indices of silicon and silicon oxide used in our simulations are *nSi* ¼ 3*:*45, *nSiO*<sup>2</sup> ¼ 1*:*45. The field profiles of the fundamental mode and the first-order mode of the multimode waveguide are shown in **Figure 2(c)** and **(d)**.

compared with structures based on directional couplers.

of fabrication with the current CMOS circuit.

coupling coefficient.

mable in optoelectronics.

**2.1 Device structure**

**50**

four ports is shown in **Figure 1**.

*(a) SOI waveguide structure, (b) field profile of the single-mode SOI waveguide, (c) fundamental mode of the multimode waveguide, and (d) the first-order mode of the multimode waveguide.*

In this structure, we use a bent waveguide to connect input port 3 to port 4 as a ring resonator waveguide. Because port 3 is very near to port 4, the bent waveguide radius is relatively small. Therefore, our structure can provide a very high free spectral range (FSR), which is suitable for high-speed communications.

In the next section, we show that our structure can act like a microring resonator. In order to control the critical coupling, we use graphene integrated with silicon waveguide. Graphene can be incorporated into silicon to implement graphenesilicon waveguide. The length of the graphene waveguide is *Larm*. The cross-section view of the graphene-silicon waveguide is shown in **Figure 3(a)**. The GSW has a monolayer graphene sheet of 340 nm on top of a silicon waveguide, separated from it by a thin Al2O3 layer. Graphene, Al2O3, and silicon together formed a capacitor structure, which was the basic block of the graphene modulator and phase shifter [25]. The refractive index of Al2O3 used in our simulations is 1.6 at the operating wavelength of 1550 nm.

For example, the field profile of the waveguide with a chemical potential *μ<sup>c</sup>* ¼ 0*:*45*eV* is shown in **Figure 3(b)**.

In a multimode waveguide, the information of the image position in the x direction and phases of the output images is very important. We need to know where the multi-images appear in order to design output waveguides to capture the optical output. Furthermore, phase information of the spot images or output images is important for such devices as MMI switch. It can be shown that the field in the multimode region will be of the form [12]

$$\mathbf{f}(\mathbf{x}, \mathbf{L}\_{\text{MMI}}) = \frac{1}{\sqrt{N}} \sum\_{p=0}^{N-1} f\_{\text{in}}(\mathbf{x} - \mathbf{x}\_p) \exp\left(-j\rho\_p\right) \tag{1}$$

We showed that the characteristics of an MMI device can be described by a transfer matrix [2]. This transfer matrix is a very useful tool for analyzing cascaded MMI structures. Phase *ϕij* is associated with imaging an input i to an output j in an MMI coupler. These phases *ϕij* form a matrix *S*4*<sup>x</sup>*4, with i representing the row number and j representing the column number. A single 4 � 4 MMI coupler at a

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon…*

**<sup>S</sup>**4*x*<sup>4</sup> <sup>¼</sup> <sup>1</sup> 2

*DOI: http://dx.doi.org/10.5772/intechopen.92210*

*\*Eout* <sup>¼</sup>

*E*in,1 *<sup>E</sup>*in,4 � � <sup>¼</sup> **<sup>M</sup>**

2

waveguide and can be calculated by [28]

*Eout*,1 *Eout*,2 *Eout*,3 *Eout*,4 1

*E*in,2 *<sup>E</sup>*in,3 � � <sup>¼</sup> *<sup>e</sup> <sup>j</sup>*

2

<sup>Δ</sup>*<sup>φ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> λ*

BPM [29]. We show that the optimal length is found to be 214 μm.

graphene-silicon waveguide with the length of *Larm* and the silicon on insulator

The phase difference Δ*φ* can be controlled by applying a voltage *Vg* to the graphene sheet. The field propagation of the multimode waveguide for input ports 1 and 2 is shown in **Figure 4**. The optimal length of the MMI is calculated by the 3D-

*Field propagation of 4* � *4 MMI coupler: (a) field propagation, in 1, and (b) field propagation, in 2.*

CCCA <sup>¼</sup> **<sup>S</sup>**4*x*<sup>4</sup>

0

BBB@

� �, and *<sup>κ</sup>* <sup>¼</sup> sin <sup>Δ</sup>*<sup>φ</sup>*

<sup>2</sup> is described by the following transfer matrix [26, 27]:

1 � *j* 0 01 þ *j* 0 1 � *j* 1 þ *j* 0 0 1 þ *j* 1 � *j* 0 1 þ *j* 0 01 � *j*

> *E*in,1 *E*in,2 *E*in,3 *E*in,4

1

*τ κ <sup>κ</sup>* <sup>∗</sup> �*<sup>τ</sup>* <sup>∗</sup>

� � *E*in,2

� �, Δ*φ*is the phase difference between the

Δ*neffLarm* (6)

CCCA <sup>¼</sup> **<sup>S</sup>**4*x*<sup>4</sup>

0

BBB@

where *Ein*,*<sup>i</sup>* (i = 1, 2, 3, 4) and *E*out,j (j = 1, 2, 3, 4) are complex amplitudes at input ports and output ports 1–4, respectively. From Eqs. (3) and (4), we can calculate the relations ships between the input and output amplitudes of **Figure 1** as follow:

> Δ*φ* 2

The output and input amplitudes at four ports of the 4 � 4 multimode wave-

(3)

*\*Ein* (4)

*<sup>E</sup>*in,3 � � (5)

length of *LMMI* <sup>¼</sup> <sup>3</sup>*L<sup>π</sup>*

guide can be expressed by

where *<sup>τ</sup>* <sup>¼</sup> cos <sup>Δ</sup>*<sup>φ</sup>*

**Figure 4.**

**53**

where *xp* <sup>¼</sup> *<sup>b</sup>*ð Þ <sup>2</sup>*<sup>p</sup>* � *<sup>N</sup> WMMI <sup>N</sup>* , *<sup>φ</sup><sup>p</sup>* <sup>¼</sup> *b N*ð Þ � *<sup>p</sup> <sup>p</sup><sup>π</sup> <sup>N</sup>* ; *fin*ð Þ *x* describes the field profile at the input of the multimode region; *xp* and *φ<sup>p</sup>* describe the positions and phases, respectively, of N self-images at that output of the multimode waveguide; p denotes the output image number; and b describes a multiple of the imaging length. For short device, we choose b = 1.

Consider a 4 � 4 multimode waveguide with the length of *<sup>L</sup>* <sup>¼</sup> *LMMI* <sup>¼</sup> <sup>3</sup>*L<sup>π</sup>* 2 , where *<sup>L</sup><sup>π</sup>* <sup>¼</sup> *<sup>π</sup> β*0�*β*<sup>1</sup> is the beat length of the MMI and *β*0, *β*<sup>1</sup> are the propagation constants of the fundamental and first-order modes supported by the multimode waveguide with a width of *WMMI*. The phases associated with the images from input i to output j can be presented by

$$\rho\_{\vec{\eta}} = -\frac{\pi}{2}(-\mathbf{1})^{i+j+4} + \frac{\pi}{16} \left[ i + j - i^2 - j^2 + (-\mathbf{1})^{i+j+4} \left( 2\mathbf{i}\mathbf{j} - \mathbf{i} - \mathbf{j} + \frac{\mathbf{1}}{2} \right) \right] \tag{2}$$

**Figure 3.** *(a) Graphene-silicon waveguide structure and (b) field profile with a chemical potential of μ<sup>c</sup>* ¼ 0*:*45*eV.*

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon… DOI: http://dx.doi.org/10.5772/intechopen.92210*

We showed that the characteristics of an MMI device can be described by a transfer matrix [2]. This transfer matrix is a very useful tool for analyzing cascaded MMI structures. Phase *ϕij* is associated with imaging an input i to an output j in an MMI coupler. These phases *ϕij* form a matrix *S*4*<sup>x</sup>*4, with i representing the row number and j representing the column number. A single 4 � 4 MMI coupler at a length of *LMMI* <sup>¼</sup> <sup>3</sup>*L<sup>π</sup>* <sup>2</sup> is described by the following transfer matrix [26, 27]:

$$\mathbf{S}\_{4\times 4} = \frac{1}{2} \begin{bmatrix} \mathbf{1} - j & \mathbf{0} & \mathbf{0} & \mathbf{1} + j \\ \mathbf{0} & \mathbf{1} - j & \mathbf{1} + j & \mathbf{0} \\ \mathbf{0} & \mathbf{1} + j & \mathbf{1} - j & \mathbf{0} \\ \mathbf{1} + j & \mathbf{0} & \mathbf{0} & \mathbf{1} - j \end{bmatrix} \tag{3}$$

The output and input amplitudes at four ports of the 4 � 4 multimode waveguide can be expressed by

$$
\overrightarrow{E\_{out}} = \begin{pmatrix} E\_{out,1} \\ E\_{out,2} \\ E\_{out,3} \\ E\_{out,4} \end{pmatrix} = \mathbf{S}\_{4 \times 4} \begin{pmatrix} E\_{\text{in},1} \\ E\_{\text{in},2} \\ E\_{\text{in},3} \\ E\_{\text{in},4} \end{pmatrix} = \mathbf{S}\_{4 \times 4} \overrightarrow{E\_{in}} \tag{4}
$$

where *Ein*,*<sup>i</sup>* (i = 1, 2, 3, 4) and *E*out,j (j = 1, 2, 3, 4) are complex amplitudes at input ports and output ports 1–4, respectively. From Eqs. (3) and (4), we can calculate the relations ships between the input and output amplitudes of **Figure 1** as follow:

$$
\begin{pmatrix} E\_{\text{in,1}} \\ E\_{\text{in,4}} \end{pmatrix} = \mathbf{M} \begin{pmatrix} E\_{\text{in,2}} \\ E\_{\text{in,3}} \end{pmatrix} = e^{\frac{\mu\_{\text{2}}}{2}} \begin{bmatrix} \tau & \kappa \\ \kappa^\* & -\tau^\* \end{bmatrix} \begin{pmatrix} E\_{\text{in,2}} \\ E\_{\text{in,3}} \end{pmatrix} \tag{5}
$$

where *<sup>τ</sup>* <sup>¼</sup> cos <sup>Δ</sup>*<sup>φ</sup>* 2 � �, and *<sup>κ</sup>* <sup>¼</sup> sin <sup>Δ</sup>*<sup>φ</sup>* 2 � �, Δ*φ*is the phase difference between the graphene-silicon waveguide with the length of *Larm* and the silicon on insulator waveguide and can be calculated by [28]

$$
\Delta \rho = \frac{2\pi}{\lambda} \Delta n\_{\rm eff} L\_{arm} \tag{6}
$$

The phase difference Δ*φ* can be controlled by applying a voltage *Vg* to the graphene sheet. The field propagation of the multimode waveguide for input ports 1 and 2 is shown in **Figure 4**. The optimal length of the MMI is calculated by the 3D-BPM [29]. We show that the optimal length is found to be 214 μm.

**Figure 4.** *Field propagation of 4* � *4 MMI coupler: (a) field propagation, in 1, and (b) field propagation, in 2.*

waveguide. Graphene can be incorporated into silicon to implement graphenesilicon waveguide. The length of the graphene waveguide is *Larm*. The cross-section view of the graphene-silicon waveguide is shown in **Figure 3(a)**. The GSW has a monolayer graphene sheet of 340 nm on top of a silicon waveguide, separated from it by a thin Al2O3 layer. Graphene, Al2O3, and silicon together formed a capacitor structure, which was the basic block of the graphene modulator and phase shifter [25]. The refractive index of Al2O3 used in our simulations is 1.6 at the operating

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

For example, the field profile of the waveguide with a chemical potential

In a multimode waveguide, the information of the image position in the x direction and phases of the output images is very important. We need to know where the multi-images appear in order to design output waveguides to capture the optical output. Furthermore, phase information of the spot images or output images is important for such devices as MMI switch. It can be shown that the field in the

> 1 ffiffiffiffi *N* <sup>p</sup> <sup>X</sup> *N*�1

*p*¼0

the input of the multimode region; *xp* and *φ<sup>p</sup>* describe the positions and phases, respectively, of N self-images at that output of the multimode waveguide; p denotes the output image number; and b describes a multiple of the imaging length. For

Consider a 4 � 4 multimode waveguide with the length of *<sup>L</sup>* <sup>¼</sup> *LMMI* <sup>¼</sup> <sup>3</sup>*L<sup>π</sup>*

constants of the fundamental and first-order modes supported by the multimode waveguide with a width of *WMMI*. The phases associated with the images from

<sup>2</sup> � *<sup>j</sup>*

*(a) Graphene-silicon waveguide structure and (b) field profile with a chemical potential of μ<sup>c</sup>* ¼ 0*:*45*eV.*

*<sup>N</sup>* , *<sup>φ</sup><sup>p</sup>* <sup>¼</sup> *b N*ð Þ � *<sup>p</sup> <sup>p</sup><sup>π</sup>*

*fin x* � *xp*

is the beat length of the MMI and *β*0, *β*<sup>1</sup> are the propagation

� � exp �*jφ<sup>p</sup>*

<sup>2</sup> þ �ð Þ<sup>1</sup> *<sup>i</sup>*þ*j*þ<sup>4</sup> 2ij � <sup>i</sup> � <sup>j</sup> <sup>þ</sup>

� � � �

� �

*<sup>N</sup>* ; *fin*ð Þ *x* describes the field profile at

1 2 (1)

2 ,

(2)

wavelength of 1550 nm.

*μ<sup>c</sup>* ¼ 0*:*45*eV* is shown in **Figure 3(b)**.

multimode region will be of the form [12]

where *xp* <sup>¼</sup> *<sup>b</sup>*ð Þ <sup>2</sup>*<sup>p</sup>* � *<sup>N</sup> WMMI*

short device, we choose b = 1.

*β*0�*β*<sup>1</sup>

input i to output j can be presented by

ð Þ �<sup>1</sup> *<sup>i</sup>*þ*j*þ<sup>4</sup> <sup>þ</sup> *<sup>π</sup>*

<sup>16</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>j</sup>* � *<sup>i</sup>*

where *<sup>L</sup><sup>π</sup>* <sup>¼</sup> *<sup>π</sup>*

*<sup>φ</sup>ij* ¼ � *<sup>π</sup>* 2

**Figure 3.**

**52**

fð Þ¼ *x*, L*MMI*

*<sup>σ</sup>*intrað Þ¼ *<sup>ω</sup> ie*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92210*

graphene.

**Figure 6.**

**55**

*Effective refractive index of the GSW waveguide.*

the chemical potential:

**2.3 Critical coupling control**

*π*ℏð Þ *ω* þ *i*2Γ

*<sup>σ</sup>*interð Þ¼� *<sup>ω</sup> ie*<sup>2</sup>

*μ<sup>c</sup>* V*<sup>g</sup>* � � � � � *μc*

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon…*

where e is electron charge, ℏ is the angular Planck constant, *kB* is the Boltzmann constant, T is the temperature, *<sup>μ</sup><sup>c</sup>* is the Fermi level or chemical potential, <sup>Γ</sup> <sup>¼</sup> *eV*<sup>2</sup>

the electron collision rate, *μ* is electron mobility, and *VF* is the Fermi velocity in

The dielectric constant of a graphene layer can be calculated by [21, 22]

*ε<sup>g</sup>* ð Þ¼ *ω* 1 þ

� ¼ ℏ*VF*

where *V*<sup>0</sup> is the offset voltage from zero caused by natural doping.

It is shown that the normalized transmission T through the device can be switched from unity to zero at the condition of critical coupling, given by *α* ¼ j j *τ* . The control of the phase shift Δ*φ*, so the condition of the critical coupling is met, can be achieved through applying voltage to the graphene sheet. The effective index of the graphene-silicon waveguide calculated at different chemical potentials by FDM method is shown in **Figure 6**. We see that for low-loss waveguide, the

The refractive index of the graphene layer sheet can be changed by providing the applied voltage *Vg* to the graphene sheet. It is because it will change the value of

*kBT* <sup>þ</sup> 2 ln e�ð Þ *<sup>μ</sup>c=*k*B*<sup>T</sup> <sup>þ</sup> <sup>1</sup>

2 *μ<sup>c</sup>* j j þ ð Þ *ω* � 2*i*Γ ℏ

<sup>4</sup>*π*<sup>ℏ</sup> ln <sup>2</sup> *<sup>μ</sup><sup>c</sup>* j j � ð Þ *<sup>ω</sup>* � <sup>2</sup>*i*<sup>Γ</sup> <sup>ℏ</sup>

*iσ ω*ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π η* V*<sup>g</sup>* � V0 � � � � � � <sup>q</sup>

� � � � (10)

� � (11)

*ωε*0<sup>Δ</sup> (12)

*F μμ<sup>c</sup>* is

(13)

**Figure 5.** *Refractive index of graphene sheet.*

The light propagation through the resonator is characterized by a round-trip transmission *Ein*,3 <sup>¼</sup> *<sup>α</sup>* exp jð Þ*<sup>θ</sup> Ein*,4, where *<sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>λ</sup> neffLR* is the round-trip phase, *α* is the loss factor, *neff* is the effective refractive index of the SOI single-mode waveguide, and *LR* is the ring resonator circumference. The normalized transmitted power of the device can be calculated by

$$T = \left| \frac{E\_{in,2}}{E\_{in,1}} \right|^2 = \frac{a^2 - \cos^2(\Delta\rho/2) - 2a|\cos\left(\Delta\rho/2\right)|\cos\left(\theta\right)}{1 + a^2\cos^2(\Delta\rho/2) - 2a|\cos\left(\Delta\rho/2\right)|\cos\left(\theta\right)}\tag{7}$$

At resonance wavelengths when *θ* ¼ 2*mπ*, m ¼ 1, 2, 3, … , the normalized power transmission is

$$T = \left| \frac{E\_{\text{in},2}}{E\_{\text{in},1}} \right|^2 = \frac{|a - |\cos\left(\Delta\rho/2\right)| |^2}{|\mathbf{1} - a|\cos\left(\Delta\rho/2\right)| |^2} \tag{8}$$

#### **2.2 Graphene-silicon waveguide**

The presence of the graphene layer changes the propagation characteristics of the guided modes, and these can be controlled and reconfigured, changing the chemical potential by means of applying a suitable voltage *Vg*. The real and image parts of the refractive index of graphene with different chemical potentials are shown in **Figure 5** [30].

Graphene has optical properties due to its band structure that provides both intraband and interband transitions. Both types of the transitions contribute to the material conductivity expressed by

$$
\sigma(o) = \sigma\_{\text{intra}}(o) + \sigma\_{\text{inter}}(o) \tag{9}
$$

where *σ*intrað Þ *ω* and *σ*interð Þ *ω* are the intraband and interband conductivities, which can be calculated by the Kubo's theory:

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon… DOI: http://dx.doi.org/10.5772/intechopen.92210*

$$\sigma\_{\rm intra}(\boldsymbol{\alpha}) = \frac{i e^2}{\pi \hbar (\boldsymbol{\alpha} + i \mathbf{2} \boldsymbol{\Gamma})} \left[ \frac{\mu\_c}{k\_B T} + 2 \ln \left( \mathbf{e}^{-(\mu\_c/k\_B \mathbf{T})} + \mathbf{1} \right) \right] \tag{10}$$

$$\sigma\_{\text{inter}}(\boldsymbol{\omega}) = -\frac{ie^2}{4\pi\hbar} \ln\left(\frac{2|\mu\_c| - (\boldsymbol{\omega} - 2i\Gamma)\hbar}{2|\mu\_c| + (\boldsymbol{\omega} - 2i\Gamma)\hbar}\right) \tag{11}$$

where e is electron charge, ℏ is the angular Planck constant, *kB* is the Boltzmann constant, T is the temperature, *<sup>μ</sup><sup>c</sup>* is the Fermi level or chemical potential, <sup>Γ</sup> <sup>¼</sup> *eV*<sup>2</sup> *F μμ<sup>c</sup>* is the electron collision rate, *μ* is electron mobility, and *VF* is the Fermi velocity in graphene.

The dielectric constant of a graphene layer can be calculated by [21, 22]

$$\varepsilon\_{\rm g}(o) = 1 + \frac{i\sigma(o)}{o\varepsilon\_0 \Delta} \tag{12}$$

The refractive index of the graphene layer sheet can be changed by providing the applied voltage *Vg* to the graphene sheet. It is because it will change the value of the chemical potential:

$$\left|\mu\_{\rm c}(\mathbf{V}\_{\rm g})\right| = \hbar V\_F \sqrt{\pi \left|\eta \left(\mathbf{V}\_{\rm g} - \mathbf{V}\_0\right)\right|}\tag{13}$$

where *V*<sup>0</sup> is the offset voltage from zero caused by natural doping.

#### **2.3 Critical coupling control**

The light propagation through the resonator is characterized by a round-trip

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

the loss factor, *neff* is the effective refractive index of the SOI single-mode waveguide, and *LR* is the ring resonator circumference. The normalized transmitted

<sup>¼</sup> *<sup>α</sup>*<sup>2</sup> � cos <sup>2</sup>ð Þ� <sup>Δ</sup>*φ=*<sup>2</sup> <sup>2</sup>*α*j j cosð Þ <sup>Δ</sup>*φ=*<sup>2</sup> cosð Þ*<sup>θ</sup>*

<sup>¼</sup> j j *<sup>α</sup>* � j j cosð Þ <sup>Δ</sup>*φ=*<sup>2</sup> <sup>2</sup>

At resonance wavelengths when *θ* ¼ 2*mπ*, m ¼ 1, 2, 3, … , the normalized power

The presence of the graphene layer changes the propagation characteristics of the guided modes, and these can be controlled and reconfigured, changing the chemical potential by means of applying a suitable voltage *Vg*. The real and image parts of the refractive index of graphene with different chemical potentials are

Graphene has optical properties due to its band structure that provides both intraband and interband transitions. Both types of the transitions contribute to the

where *σ*intrað Þ *ω* and *σ*interð Þ *ω* are the intraband and interband conductivities,

<sup>1</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> cos <sup>2</sup>ð Þ� <sup>Δ</sup>*φ=*<sup>2</sup> <sup>2</sup>*α*j j cosð Þ <sup>Δ</sup>*φ=*<sup>2</sup> cosð Þ*<sup>θ</sup>* (7)

*σ ω*ð Þ¼ *σ*intrað Þþ *ω σ*interð Þ *ω* (9)

j j <sup>1</sup> � *<sup>α</sup>*j j cosð Þ <sup>Δ</sup>*φ=*<sup>2</sup> <sup>2</sup> (8)

*<sup>λ</sup> neffLR* is the round-trip phase, *α* is

transmission *Ein*,3 <sup>¼</sup> *<sup>α</sup>* exp jð Þ*<sup>θ</sup> Ein*,4, where *<sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

 

2

*<sup>T</sup>* <sup>¼</sup> *Ein*,2 *Ein*,1 

 

2

power of the device can be calculated by

*<sup>T</sup>* <sup>¼</sup> *Ein*,2 *Ein*,1 

**2.2 Graphene-silicon waveguide**

material conductivity expressed by

which can be calculated by the Kubo's theory:

shown in **Figure 5** [30].

**54**

transmission is

**Figure 5.**

*Refractive index of graphene sheet.*

It is shown that the normalized transmission T through the device can be switched from unity to zero at the condition of critical coupling, given by *α* ¼ j j *τ* . The control of the phase shift Δ*φ*, so the condition of the critical coupling is met, can be achieved through applying voltage to the graphene sheet. The effective index of the graphene-silicon waveguide calculated at different chemical potentials by FDM method is shown in **Figure 6**. We see that for low-loss waveguide, the

**Figure 6.** *Effective refractive index of the GSW waveguide.*

chemical potential should be larger than *μ<sup>c</sup>* ¼ 0*:*4*eV*. By changing the chemical potential, the transmission and coupling coefficients j j *τ* , j j *κ* can be changed as shown in **Figure 7**. The simulations show that we can get full control of the coupling coefficient from zero to unity by changing the chemical potential from *μ<sup>c</sup>* ¼ 0*:*4*eV*

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon…*

The transmissions of the device at different chemical potentials and loss factors are shown in **Figure 8**. The simulations have two very important features which are the key for most of the proposed applications: (1) The transmitted power is zero at a value of critical coupling, and (2) for high-quality factor, the portion of the curve to the right of the critical coupling point is steep. Small changes of the phase shifter can control the transmitted power and switch between unity and zero [17]. This chapter shows that we can achieve high-speed devices based on our proposed

Some other performance parameters of the microring resonator are finesse, Q-factor, resonance width, and bandwidth. These are all terms that are mainly related to the full width at half of the maximum (FWHM) of the transmission. The quality factor Q of the microring resonator of the structure in **Figure 1** can be derived as [3]

> ffiffiffiffiffi *ατ* p

> > ffiffiffiffiffi *ατ* p

<sup>1</sup> � *ατ* (14)

<sup>1</sup> � *ατ* (15)

*<sup>Q</sup>* <sup>¼</sup> *<sup>π</sup>NgLR λ*

defined and calculated for the single- and add-drop microring resonators by

*<sup>F</sup>* <sup>¼</sup> *FSR* Δ*λFWHM*

Another important parameter for microring resonators is the finesse F, which is

¼ *π*

to *μ<sup>c</sup>* ¼ 0*:*58*eV*.

*DOI: http://dx.doi.org/10.5772/intechopen.92210*

microring resonator.

**Figure 9.**

**57**

*Finesse and quality factor at different chemical potentials.*

**Figure 7.** *Transmission and coupling coefficients of the resonator with different chemical potentials.*

**Figure 8.** *Transmissions of the microring resonator with different chemical potential and loss factor.*

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon… DOI: http://dx.doi.org/10.5772/intechopen.92210*

chemical potential should be larger than *μ<sup>c</sup>* ¼ 0*:*4*eV*. By changing the chemical potential, the transmission and coupling coefficients j j *τ* , j j *κ* can be changed as shown in **Figure 7**. The simulations show that we can get full control of the coupling coefficient from zero to unity by changing the chemical potential from *μ<sup>c</sup>* ¼ 0*:*4*eV* to *μ<sup>c</sup>* ¼ 0*:*58*eV*.

The transmissions of the device at different chemical potentials and loss factors are shown in **Figure 8**. The simulations have two very important features which are the key for most of the proposed applications: (1) The transmitted power is zero at a value of critical coupling, and (2) for high-quality factor, the portion of the curve to the right of the critical coupling point is steep. Small changes of the phase shifter can control the transmitted power and switch between unity and zero [17]. This chapter shows that we can achieve high-speed devices based on our proposed microring resonator.

Some other performance parameters of the microring resonator are finesse, Q-factor, resonance width, and bandwidth. These are all terms that are mainly related to the full width at half of the maximum (FWHM) of the transmission. The quality factor Q of the microring resonator of the structure in **Figure 1** can be derived as [3]

$$Q = \frac{\pi N\_{\text{g}} L\_{\text{R}}}{\lambda} \frac{\sqrt{a\pi}}{1 - a\pi} \tag{14}$$

Another important parameter for microring resonators is the finesse F, which is defined and calculated for the single- and add-drop microring resonators by

¼ *π*

ffiffiffiffiffi *ατ* p

*<sup>F</sup>* <sup>¼</sup> *FSR* Δ*λFWHM*

$$F = \frac{\sum\_{\mathbf{i}\_{\text{cell}}} \mathbf{r}\_{\mathbf{i}}}{\Delta \lambda\_{\text{FWHM}}} = \pi \frac{\mathbf{v}\_{\mathbf{i}\_{\text{cell}}}}{\mathbf{1} - \alpha \tau} \tag{15}$$

$$\bigwedge\_{\mathbf{i}\_{\text{cell}}} \mathbf{r}\_{\mathbf{i}\_{\text{cell}}} = \underbrace{\mathbf{r}\_{\mathbf{i}\_{\text{cell}}} \mathbf{r}\_{\mathbf{i}\_{\text{cell}}}^{\text{\text{\textquotedblleft}10^{4}}}\_{\mathbf{i}\_{\text{cell}}}}\_{\mathbf{i}\_{\text{cell}}}$$

**Figure 9.** *Finesse and quality factor at different chemical potentials.*

**Figure 7.**

**Figure 8.**

**56**

*Transmission and coupling coefficients of the resonator with different chemical potentials.*

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

*Transmissions of the microring resonator with different chemical potential and loss factor.*

accurate simulations [31]. The FDTD simulations for the proposed microring resonator with chemical potential of 0.45 and 0.42 eV are shown in **Figure 11(a)** and **(b)**. The simulations show that the device operation has a good agreement with our

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon…*

We presented a new microring resonator based on only one multimode waveguide. The critical control of the microring resonator can be achieved by using graphene-silicon waveguide. The proposed device has all characteristics of a traditional microring resonator. Some important parameters of the proposed device such as finesse, quality factor, etc. are also presented in this chapter. The device operation has been verified by using the FDTD. This microring resonator structure is very compact and can be useful for further applications in optical switching, filtering,

This research is funded by the Ministry of Natural Resources and Environment

International School (VNU-IS), Vietnam National University (VNU), Hanoi,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

prediction by analytical theory.

*DOI: http://dx.doi.org/10.5772/intechopen.92210*

**3. Conclusions**

and sensing.

**Acknowledgement**

**Author details**

Vietnam

**59**

Trung-Thanh Le\* and Duy-Tien Le

provided the original work is properly cited.

\*Address all correspondence to: thanh.le@vnu.edu.vn

of Vietnam under the project BĐKH.30/16-20.

where Δ*λFWHM* is the resonance full width at half maximum and FSR is the free spectral range. The free spectral range is the distance between two peaks on a wavelength scale. By differentiating the equation *φ* ¼ *βLR*, we get *FSR* <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup> *ngL*, where the group index *ng* <sup>¼</sup> *neff* � *<sup>λ</sup> dneff <sup>d</sup><sup>λ</sup>* .

**Figure 9** shows the finesse and quality factor with different chemical potential at a radius of 5*μm*. We see that a maximum finesse and quality factor can be achieved at a chemical potential of *μ<sup>c</sup>* ¼ 0*:*57*eV*.

The normalized transmissions of the propose microring resonator in **Figure 1** at microring radii of 5*μm* and 50*μm* are shown in **Figure 10**. Here we assume that the chemical potential is *μ<sup>c</sup>* ¼ 0*:*45*eV*. The simulations show that the exact characteristics of a single microring resonator can be achieved.

Finally, we use FDTD method to simulate the proposed microring resonator based on multimode waveguide. In our FDTD simulations, we take into account the wavelength dispersion of the silicon waveguide. A light pulse of 15 fs pulse width is launched from the input to investigate the transmission characteristics of the device. The grid sizes Δx = Δy = Δz = 20 nm are chosen in our simulations for

**Figure 11.** *Optical field propagation through the coupler for input signal presented at (a) port 1 and (b) port 2.*

*High FSR and Critical Coupling Control of Microring Resonator Based on Graphene-Silicon… DOI: http://dx.doi.org/10.5772/intechopen.92210*

accurate simulations [31]. The FDTD simulations for the proposed microring resonator with chemical potential of 0.45 and 0.42 eV are shown in **Figure 11(a)** and **(b)**. The simulations show that the device operation has a good agreement with our prediction by analytical theory.
